Class_21 - Introduction to Multiple Regression Analysis You...

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Introduction to Multiple Regression Analysis
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You will recall that the general linear model used in least squares regression is: Y i = α + bX i + ε I where b is the regression coefficient describing the average change in Y per one unit increase (or decrease) in X, α is the Y-intercept (the point where the line of best fit crosses the y-axis), and ε i is the error term or residual (the difference between the actual value of Y for a given value of X and the value of Y predicted by the regression model for that same value of X). In other words, the regression coefficient describes the influence of X on Y. But how can we tell if this influence is a causal influence?
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To answer this question, we need to satisfy the three criteria for labeling X the cause of Y: (1) that there is covariation between X and Y; (2) that X precedes Y in time; and (3) that nothing but X could be the cause of Y.
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(1) We will know that X and Y covary if the regression coefficient is statistically significant . This means that β (the "true" relationship in the universe, i.e., in general) is probably not 0.0. (Remember, b = 0.0 means that X and Y are statistically independent. Therefore, X could not be the cause of Y.) (2) We will know that X precedes Y in time if we have chosen our variables carefully. (There is no statistical test for time order; it must be dealt with through measurement, research design, etc.)
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(3) How can we tell if something other than X could be the cause of Y?; as before, by introducing control variables . We saw how this was done with discrete variables used to create zero-order and first-order partial contingency tables. In the case of regression analysis, statistical control is achieved by adding control variables to our (linear) regression model.
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This transforms simple regression into multiple regression analysis . The multiple regression model looks like this: Y i = α + b 1 X 1i + b 2 X 2i + b 3 X 3i + ε i We still have a residual and a Y-intercept. However, by introducing two additional variables in our regression model on the right-hand side (they are sometimes called "right-side" variables), we have changed the relationship from a (sort of zero-order) bivariate X and Y association to a multi-way relationship with two control variables, X 2 and X 3 . We therefore have three regression coefficients, b 1 , b 2 , and b 3 .
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The central point is that, when we solve for the values of these constants, the value of b 1 (the coefficient for our presumed cause, X 1 ) now has been automatically adjusted for whatever influence the control variables, X 2 and X 3 , have on Y. This adjustment occurs mathematically in the solution of simultaneous equations involving four unknowns, α , b 1 , b 2 , and b 3 . In other words, instead of describing the gross influence of X 1 on Y as in the simple regression case, in the multiple regression case this coefficient describes the net influence of X 1 on Y, that is, net of the effects of X 2 and X 3 . This is statistical control at its best and is the way we answer the question of non- spuriousness.
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How do we interpret the results?
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